Page:Proceedings of the Royal Society of London Vol 1.djvu/59

Rh Some observations are lastly added concerning the nature and situation of the ciliary processes in various animals; also on the nature of the marsupium nigrum of birds, and the horseshoe-like appearance in the choroid of fishes; both which have improperly been termed muscular,-the former being a mere duplicature of a membrane which may be unfolded; and in the latter the whole mass being evidently of an uniform texture, the fibrous appearance which has misled some former observers being the effect of transverse fis- sures, or cracks, which may easily be mistaken for filaments.

The lecture concludes with a few observations on the bony scales of the eyes of birds, to which the author denies any concern in changing the focus of the eye; and on a cavity observable in the eyes of some insects which has been supposed to be in some measure subservient to this purpose.

The object of this paper is to show, that we may be assured of the justness and accuracy of conclusions obtained by means of imaginary quantities, without verifying such conclusions by separate investiga- tions, or without inferring their truth from analogy. In the first part the author premises at some length certain arguments, to show that the operations with impossible quantities must have a logic equally strict and certain with the logic that appertains to real quantities, and that the aid obtained by these quantities would be perfectly use- less if such conclusions rested only on the frail basis of analogy.

The author proceeds next to show that operations with imaginary quantities are by no means mechanical, but that they are conducted according to the rules of strict and rigorous logic; and that, although strictly speaking no proposition concerning them can be true or false, yet, after the demonstrations of certain formule for real quantities, demonstrations with impossible quantities may be legitimately and logically conducted. The series, for instance, for the development of an exponential, when the exponent is an impossible quantity, can never, independently of certain arbitrary assumptions, be duly esta- blished; and yet, when the exponent is the sign of a real quantity, the formula for the development may be rigorously proved. With regard to demonstration, it is shown, as in the case of real quantities, it actually proceeds by a series of transformation, each proved to be the same as the foregoing, not by any arguments grounded on the properties of real quantities, but by reference to the forms certain abridged symbols are made to represent, and to the nature of the operations directed to be performed with certain signs.

After thus establishing the principle by which operations with imaginary characters are regulated, the author shows its efficacy and